A164117 -id:A164117 - cp's OEIS frontend

A177704 Period 4: repeat [1, 1, 1, 2].

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1

Offset: 0

Klaus Brockhaus, May 11 2010

Continued fraction expansion of (3 + 2*sqrt(6))/5.
Decimal expansion of 1112/9999.
a(n) = A164115(n + 1) = (-1)^(n + 1) * A164117(n + 1) = A138191(n + 3) = A107453(n + 5).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
David Nacin, Van der Laan Sequences and a Conjecture on Padovan Numbers, J. Int. Seq., Vol. 26 (2023), Article 23.1.2.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A001068, A107453, A138191, A164115, A164117, A177705, A219977, A236398.

Magma
```
&cat[ [1, 1, 1, 2]: k in [1..27] ];
```

A177704:=n->floor((n+1)*5/4) - floor(n*5/4): seq(A177704(n), n=0..100); # Wesley Ivan Hurt, Jun 15 2016

Table[Floor[(n + 1)*5/4] - Floor[n*5/4], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 15 2016 *)
LinearRecurrence[{0, 0, 0, 1}, {1, 1, 1, 2}, 100] (* Vincenzo Librandi, Jun 16 2016 *)

a(n) = if(n%4==3, 2, 1) \\ Felix Fröhlich, Jun 15 2016

a(n) = (5-(-1)^n + i*i^n-i*(-i)^n)/4 where i = sqrt(-1).
a(n) = a(n-4) for n > 3; a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 2.
G.f.: (1+x+x^2+2*x^3)/(1-x^4).
a(n) = 1 + (1-(-1)^n) * (1+i^(n+1))/4 where i = sqrt(-1). - Bruno Berselli, Apr 05 2011
a(n) = 5/4 - sin(Pi*n/2)/2 - (-1)^n/4. - R. J. Mathar, Oct 08 2011
a(n) = floor((n+1)*5/4) - floor(n*5/4). - Hailey R. Olafson, Jul 23 2014
From Wesley Ivan Hurt, Jun 15 2016: (Start)
a(n+3) - a(n+2) = A219977(n).
Sum_{i=0..n-1} a(i) = A001068(n). (End)
E.g.f.: (-sin(x) + 3*sinh(x) + 2*cosh(x))/2. - Ilya Gutkovskiy, Jun 15 2016

A164115 Expansion of (1 - x^5) / ((1 - x) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2

Offset: 0

Michael Somos, Aug 10 2009

The sequence A107453 has the same terms but different offset.
Convolution inverse of A164116.
Decimal expansion of 11111/99990. - Elmo R. Oliveira, Feb 18 2024

			1 + x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + x^10 + ...

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A107453, A138191, A164116, A164117.

m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x+x^2+x^3+x^4)/(1-x^4))); // G. C. Greubel, Sep 22 2018

CoefficientList[Series[(1+x+x^2+x^3+x^4)/(1-x^4), {x, 0, 100}], x] (* G. C. Greubel, Sep 22 2018 *)
LinearRecurrence[{0,0,0,1},{1,1,1,1,2},120] (* or *) PadRight[{1},120,{2,1,1,1}] (* Harvey P. Dale, Aug 24 2019 *)

PARI
```
{a(n) = 2 - (n==0) - (n%4>0)}
```

x='x+O('x^99); Vec((1-x^5)/((1-x)*(1-x^4))) \\ Altug Alkan, Sep 23 2018

Euler transform of length-5 sequence [ 1, 0, 0, 1, -1].
a(n) is multiplicative with a(2) = 1, a(2^e) = 2 if e>1, a(p^e) = 1 if p>2.
a(n) = (-1)^n * A164117(n).
a(4*n) = 2 unless n=0. a(2*n + 1) = a(4*n + 2) = 1.
a(-n) = a(n). a(n+4) = a(n) unless n=0 or n=-4.
G.f.: (1 + x + x^2 + x^3 + x^4) / ((1+x)*(1-x)*(1+x^2)).
a(n) = A138191(n+2), n>0. - R. J. Mathar, Aug 17 2009
Dirichlet g.f. (1+1/4^s)*zeta(s). - R. J. Mathar, Feb 24 2011
a(n) = (i^n + (-i)^n + (-1)^n + 5)/4 for n > 0 where i is the imaginary unit. - Bruno Berselli, Feb 25 2011

A164118 Expansion of (1 - x^2) * (1 - x^4) * (1 - x^5) / ((1 - x) * (1 - x^10)) in powers of x.

Original entry on oeis.org

1, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1, -2, -1, 0, 0, 1, 2, 1, 0, 0, -1

Offset: 0

Michael Somos, Aug 10 2009

Index entries for linear recurrences with constant coefficients, signature (1, -1, 1, -1).

A164116(n) = (-1)^n * a(n). Convolution inverse of A164117.

CoefficientList[Series[(1 - x^4) / (1 - x + x^2 - x^3 + x^4), {x, 0, 50}], x] (* G. C. Greubel, Sep 12 2017 *)

{a(n) = -(n==0) + [2, 1, 0, 0, -1, -2, -1, 0, 0, 1][n%10 + 1]}

Euler transform of length 10 sequence [ 1, -1, 0, -1, -1, 0, 0, 0, 0, 1].
a(-n) = a(n). a(n + 5) = -a(n) unless n=0 or n=-5.
G.f.: (1 - x^4) / (1 - x + x^2 - x^3 + x^4).

cp's OEIS Frontend

A177704 Period 4: repeat [1, 1, 1, 2].

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A164115 Expansion of (1 - x^5) / ((1 - x) * (1 - x^4)) in powers of x.

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A164118 Expansion of (1 - x^2) * (1 - x^4) * (1 - x^5) / ((1 - x) * (1 - x^10)) in powers of x.

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